Music Visualization Using Dynamic Phyllotactic Patterns: A Generative Design Approach with Bézier Curve Mapping

2. 1. Music Visualization and Nature Inspiration

The field of music visualization includes a vast repertoire of widely varied techniques for the visual representation of musical data, which have been thoroughly surveyed in the latest literature (Khulusi et al., 2020). Technology for particle analysis has progressed from static spectral analysis (i.e., Fourier Transform) to real-time interactive systems (i.e., particle engines; Lima et al., 2022; Lokki et al., 2002), all of which are intended to enhance the emotional expression and cognitive involvement of music through visual media. Nevertheless, current methods like waveform mapping and particle animation are based primarily upon the direct transformation of signal processing algorithms, and as a result their visual expression is limited by the rules of parameter mapping to the mechanistic (Bale et al., 2021). Examples include color-based visualizations (Ciuha et al., 2010), image-based visualizations (Lehtiniemi and Holm, 2012; Machida and Itoh, 2011; Reeves, 1983), shape-based visualizations (Bergstrom et al., 2007; Carter-Enyi et al., 2021; Miyazaki et al., 2003; Valbom and Marcos, 2007), particle-based visualizations (Fonteles et al., 2013), and fractal representations. Another important type involves the use of mathematical models to create graphical imagery, examples of which combine graphical imagery and mathematical statistics in music visualization (W. Li & Li, 2020). While these methods, including those based on mathematical graphics, demonstrate the technical feasibility of mapping music to form, the methods too often focus on structural or statistical fidelity than on the expression of dynamic, organic, and emotionally resonant qualities. Although 3D particle systems may be responsive to musical rhythms with emitter properties (color, volume), their random burst properties do not easily create visual narratives with bio-organic qualities and restrict the user’s possibilities for deep resonance to musical emotion. This limitation highlights a crucial problem in current research: The temporal dynamics and complexity of music demand visual generating mechanisms with a higher self-organizing capacity.

In recent years, Bioinspired Design has offered a new pathway to resolve this issue. By simulating the dynamic adaptability of natural systems (e.g., phyllotactic spiral growth and fractal branching), researchers have attempted to combine the mathematical principles of biological forms (e.g., the golden angle and Fibonacci sequence) with musical features in order to generate visual expressions that integrate aesthetic harmony with emotional responsiveness (Lokki et al., 2002). However, a key distinction can be drawn within nature-inspired visualization. Most of the current achievements relate to the offline rendering of static natural patterns (e.g., fractal art; Jiang et al., 2019), which mainly mimic the aesthetic result or end form of natural structures for visual purposes. In contrast, there remains a large lacuna in the exploration of dynamic natural mechanisms (e.g., spatial competition models inherent in phyllotaxis; Prusinkiewicz et al., 2001). These mechanisms are the underlying and time-dependent processes and algorithmic rules governing the growth and formation of natural patterns. Importantly, there has thus far not been significant discussion within the field of the coupling of these dynamic mechanisms of nature and real-time musical parameters (rhythm, timbre, and emotional intensity). This gap is significant because, on the one hand, static patterns allow only a limited, predefined visual vocabulary, while on the other hand, dynamic mechanisms can create a generative “engine” able to generate coherent, adaptive, and organically evolving visuals that mirror the temporal and structural dynamics of the music itself. While organic and nature-inspired visual patterns have been successfully used in artistic performances and installations (such as the works of Universal Everything or Andres Wanner), demonstrating that they are aesthetically pleasing, the academic investigation of methods of systematically coupling the dynamic generative mechanisms of nature (as opposed to static forms alone) with real-time music remains limited. This constraint hinders the vitality and expressive dimensions of music visualization systems in interactive scenarios (e.g., live performances and virtual reality), where they might be able to create visuals that, aesthetically speaking, are not only naturalistic, but also inherently and organically responsive to musical structure in real time, much like a living system.

2. 2. Morphogenesis and Natural Pattern Generation

Morphogenesis, a central paradigm in interdisciplinary generative design, seeks to translate performance data (such as environmental constraints and functional requirements) into adaptive geometric forms by simulating the dynamic mechanisms of biological systems (e.g., plant growth; Lintermann & Deussen, 1999). The conceptual origin of this approach lies in botany: Early studies, by analyzing natural patterns like leaf arrangements and root bifurcations, revealed that biological morphogenesis emerges from the synergistic interplay between physical constraints and genetic expression (Gokmen, 2022). For instance, the spiral patterns of plant phyllotaxis have been shown to be optimal solutions to dynamic competition among growth points for spatial resources. Their mathematical representations (e.g., Vogel’s formula, Fibonacci sequence) thus provide a biomimetic foundation for computational generative algorithms (Fowler et al., 1992; Prusinkiewicz & Lindenmayer, 1990). Consequently, morphogenetic research has gradually shifted from biological explanation to design application, for which end it has been used to quantify the self-organizing rules of natural systems (e.g., stress feedback, energy minimization), from which shape search algorithms have been constructed capable of dynamically adapting to input parameters, optimizing layout efficiency, or enhancing information visualization performance (Gokmen, 2022).

Laminar patterns have become a fundamental research topic in the areas of graphics and generative design within the field of bioinspired pattern generation because of their mathematical regularity, visual harmony, and scalability. The logarithmic spirals of pinecone scales and the Fibonacci layout of sunflower seeds are not only characterized by strict geometric constraints (e.g., the golden angle), but the mechanism of their creation is inherently adaptive, following from the responses of organisms to their environment (Fowler et al., 1992; Owens et al., 2016). Nevertheless, although it is possible to reproduce static natural patterns computationally, there are still major problems when integrating them with real-time external stimuli like music.

2. 3. The Sunflower Phytotaxis Model

The seeds in a sunflower capitulum are arranged in a high-density spiral phyllotaxis pattern that adheres to the principle of maximizing nonoverlapping packing efficiency (Owens et al., 2016). Early research employed static planar layouts based on Archimedean spirals (Mathai & Davis, 1974) and Vogel’s formula (Vogel, 1979) to computationally replicate this natural phenomenon using mathematical models. Among these, Vogel’s model has become the benchmark method in computer graphics due to its parameter flexibility and computational reliability.

Furthermore, Vogel’s model has been extended to curved surfaces through geometric topological transformations, including cylindrical (Fowler et al., 1989; Prusinkiewicz & Lindenmayer, 1990) and spherical surfaces (Lintermann & Deussen, 1999; Swinbank & James Purser, 2006; Zeng & Wang, 2009) and more general surfaces of revolution (Ijiri et al., 2005). These extensions enable the mapping of natural phyllotactic patterns into three-dimensional space.

For more precise simulations of plants’ dynamic growth characteristics, the following key models have been proposed:

First, the phyllotactic dynamics framework (Prusinkiewicz et al., 2001) constructs recursive self-similar structures on uniformly growing meristems, elucidating the dynamic competition mechanism underlying directional leaf growth.

Second, growth control (Frenkel et al., 2023; Yeatts, 2004) introduces feedback through hormone concentration gradients and mechanical stress based upon the Voronoi entropy of seed configurations, which are also conceptualized as aesthetic and functionally optimal spiral patterns.

Phyllotactic patterns have a particular mathematical nature that offers new solutions in various fields (data visualization and engineering biomimetic design). In this line of research, Neumann et al. (2006) applied the Vogel model to fractal algorithms to create hierarchical structure visualization tools; Piccini et al. (2011) created 3D radial trajectories that maximize the intertwining properties of cardiac MRI images; Lyu et al. (2021) designed a microheat sink based upon phyllotactic patterns, with experiments demonstrating an 18% improvement in thermal conduction behavior; and Sarradj (2015) used a weighted optimization of microphone array layouts. Nevertheless, the current literature is mostly limited to the field of static applications.

This paper proposes a cubic Bézier weighting algorithm that employs a nonlinear mapping of musical rhythm to curvature to exploit the higher-order continuity of Bézier curves in order to overcome the mechanistic constraints of conventional particle systems (Fonteles et al., 2013).

2. 4. Rationale for Phyllotaxis Selection

Phyllotaxis (and more specifically the sunflower spiral form) was chosen in this study because it is one of the few forms whose natural characteristics include qualities that deeply correspond with the goals of expressive music visualization and, therefore, goes much beyond just reporting a research gap:

Unity of Mathematical Order and Aesthetic Appeal: Phyllotactic patterns are based on rigorous mathematical principles (such as Vogel’s formula and the golden angle). This provides a generative architecture that is able to generate visually harmonious and coherent structures. The intrinsic order resonates with the structural bases of music (e.g., tempo and meter), while the natural aesthetic quality is in line with the purpose of producing emotional responses and realizing visuals.

Inherent Dynamic and Adaptive Growth Mechanism: Importantly, phyllotaxis is not a static pattern but the product of dynamic growth through the spatial competition of primordia on the meristem. This biological mechanism is always responsive and adaptive, and is directly translated in our model into the ability to map dynamic musical features (e.g., rhythmic intensity and emotional arousal) to parameters controlling the “growth” of the pattern (e.g., divergence angle and radial scaling), whereby the visualization grows organically in connection with the music rather than being built from static fractals or prerendered patterns.

Structural Hierarchy and Self-Similarity: Phyllotactic patterns display self-similarity across scales, such that a global spiral structure is made up of smaller similar spirals. This hierarchical nature lends itself to a rich visual metaphor for understanding musical structure, whereby a piece of music is made up of movements, sections, phrases, and ultimately single notes. Consequently, our visualization is thus capable of representing both the macro level—musical form—and the micro—the rhythmic details—within a unified visual framework.

3. 1. System Architecture

To implement the music-driven dynamic phyllotaxis generation algorithm, this study employed Rhino and Grasshopper as the core development environments. Grasshopper’s node-based programming interface supports real-time parameter adjustment and dataflow visualization, facilitating the dynamic binding of mathematical modules (such as Vogel’s model and Bézier functions) with music input signals. Rhino’s NURBS geometric kernel efficiently handles the real-time rendering of complex spiral structures, meeting the dual requirements in music visualization of frame rate (≥30 fps) and graphical precision (submillimeter error). The openness of the Grasshopper component library allows the direct integration of third-party plugins (e.g., Butterfly for fluid dynamics simulation, TT Toolbox for data mapping), simplifying the cross-domain transformation process from musical features to morphological parameters. The widespread application of Rhino/Grasshopper in parametric architecture and generative art (Dean & Loy, 2020; Saadi & Yang, 2023) ensured that this research rested on a mature foundation of user testing and community support. This choice of platform facilitates the development of an intelligent interactive system, as it enables real-time parameter adjustment and visual feedback, a core requirement for interactive music information systems (Liao, 2022). Figure 1 shows the system architecture diagram of the dynamic phyllotaxis visualization system based on Rhino/Grasshopper.

Figure 1

System architecture

3. 2. Dynamic Phyllotaxis Modeling

The sunflower has a Fibonacci spiral phyllotaxis, as explained in Section 2.3 (Figures 2 and 3). Our visualization model is built on this pattern, which is ruled by the golden angle.

Figure 2

Spiral structure of the sunflower seed arrangement corresponding to Fibonacci sequence terms

Figure 3

Examples of the Fibonacci sequence observed in natural plants

Adaptation to Dynamic Visualization

The classical Vogel’s model, defined by

(where δ is the divergence angle, typically 137.5°), will lead to an equilibrium optimal-packing arrangement. To modify this model to enable real-time music visualization, we analyzed the parameters of the model to determine those that could be controlled dynamically to reflect the musical input without compromising the underlying aesthetic and structural integrity of the phyllotactic pattern.

Our adaptation reinterpreted the core parameters as functions of musical data:

Dynamic Divergence Angle (δ): For the static model δ is constant. We treated it as a time-varying parameter δ(t) modulated by rhythmic intensity and the spectral centroid for timbre. This allows the tightness and general curve of the spiral to “breathe” in response to the agencies of energy and brightness of the music. For example, an increase in rhythmic intensity can lead to an increase of δ(θ), either linear or nonlinear (using the two extremes of the Bézier mapper), which can cause the spiral to expand more rapidly.

Dynamic Radial Scaling (c): The scaling factor c in the radial factor $$ r\left(\mathrm{n}\right)=\mathrm{c}\sqrt{\mathrm{n}},$$ regulates the point density. We modulated the parameter by the amplitude envelope and pitch height. Most passages can be arranged in a way that enhances c, leading to less dense, more open patterns that imitate a reaction to some kind of energy or frequency.

Temporal Sequencing: The index n is not a point counter but a time sequence in the generation process. When new musical data are received in real-time, new points are added with coordinates (r(n,t),θ(n,t)), where the parameters c and δ have been evaluated at a given time t to create a pattern that evolves with time.

This analysis and adaptation transforms the Vogel model, a formula to generate fixed images, into a dynamic system Phyllotaxis(r(t),θ(t)) that generates visual output as a direct function of the incoming musical stream.

3. 3. Bézier Curve Mapping

This decision to use cubic Bézier curves was not arbitrary factors, but lay in the recognition that the music-based visualization demands an expressive and computationally friendly mapping mechanism. Traditional linear mappings are more likely to not capture the delicate and nonlinear character of the relationships between musical features and visual equivalents. Bézier curves allow us to address this through the following significant qualities, all of which address a vital requirement of our system:

Parametric Control and Expressive Nonlinear Mapping: The shape of a Bézier curve is determined by the locations of its control points P₁ and P₂. This provides a flexible method for developing nonlinear mappings of input (musical feature value) and output (morphological parameter). A linear change in the coordinate of a control point can result in a complicated curvilinear change to the resulting pattern. This allows the production of visually interesting and nonmechanical transformations that transcend a simple one-to-one correspondence.

Higher-Order Continuity (C²): Temporal Smoothing Music is a temporal art form. Sudden shots in the visual domain may be shocking in a song. Cubic Bézier curves are C2-continuous, meaning that their first and second derivatives (speed and acceleration of change) are everywhere continuous. This ensures smooth visual transitions despite rapid changes in the input musical data (e.g., over a drum break or a crescendo) and does not have the visual tearing-up associated with piecewise-linear or step-function mappings.

Multidimensional Parameter Coupling: The control points P₁ and P₂ are control points situated in a 2D (or 3D) space, and each x- or y-coordinate is an independent degree of freedom. This allows for the synergistic coupling of a number of musical features into one coherent morphological change. As an example, rhythm could be associated with the x-coordinate of P₁ and the timbre with its y-coordinate, which would be combined in a single control point to create a multilayered, intricate visual reaction.

The shape of a cubic four-point Bézier curve is defined by four points (P₀, P₁, P₂, P₃) in 2D or 3D space (Farin, 2002; Khanteimouri et al., 2022; Piegl, 1991). As shown in Figure 4, these points control the data range: P₀ and P₃ are fixed at coordinates (0,0) and (1,1), respectively, scaling the input/output data range from o to 1. The coordinates of P₁ and P₂ serve as controllable inputs and function as variable parameters (Böhm et al., 1984; Farin, 2002).

Figure 4

Control point configuration and curve shape of a Bézier curve

Mapping Musical Features to Control Points

The process of a concrete music-visual mapping of an abstract curve is conducted by attributing definite musical semantics to the coordinates of the control points P₁ and P₂. The rationale behind this design is as follows:

Control Point P₁ as the “Rhythm and Energy” Modulator: The location of P₁, especially the x-coordinate, mostly influences the overall curvature and tension of the phyllotactic spiral. Thus, it is mapped to features associated with rhythmic intensity and amplitude (loudness). Cross-modal similarities in this mapping are intuitively justified by the fact that increased intensity/energy is crossmodally correlated with increased or broadened visual dimensions (Spence, 2011).

Control Point P₂ as the “Pitch and Texture” Modulator: The y-coordinate of P₂ is used to control the local density and texture at the edges of the pattern. The value of this coordinate is determined by the height of the pitch and spectral centroid (brightness) of the timbre. This schema is based on well-established cross-modal correspondences, including the pitch-height effect, whereby higher pitched sounds are always linked with higher spatial positioning, and the relationship between auditory brightness and visual lightness or elevation (Spence, 2011). This strategic mapping approach employs the abstract mathematical values of the Bézier curve as valuable channels for musical expression and the direct connection of sonic qualities to visual morphology.

Figure 5 shows the mapping process of curves using the points (0,0), P₁, P₂, and (1,1) to create the Bézier curve. P₁ and P₂ are controlled through a multidimensional slider input to produce the desired curves.

Figure 5

Curve mapping configuration in the Grasshopper

In addition, this curve control algorithm was sent out in a reusable form that works with the five input variables of the control points P₁ and P₂, array start value s, pattern width w, and point count n. The relationship between these elements is indicated in Figure 6. Visual patterns naturally evolve when the values are dynamically changed based on real-time input data to create morphological flows in step with the music.

Figure 6

Custom component and input structure

The Bézier-controlled dynamic phyllotaxis model exhibits sensitive visual transformations according to the input parameters that enable pattern sculpting suitable for music visualization. This section presents a wide range of the visual patterns generated through various combinations of point count n, initial spacing s, and control point positions P₁, P₂.

Figure 7 displays spiral patterns generated by varying n and the coordinates of P₁, P₂. Both control points reside within a square domain diagonally bounded by (0,0) and (1,1), where their positions critically influence the point distribution density and curvature morphology.

Figure 7

Visual patterns by point count (n) and control point (P₁, P₂) positions

Figure 8 demonstrates the effects of varying the value of the initial array spacing s. When s > 0, a central gap emerges in the pattern, expanding proportionally with larger values of s. This adjustment breathes life into visual patterns and proves effective for visually representing beat intervals or rests (pauses) in music.

Figure 8

Visual patterns by initial spacing (s) and control point positions

However, when the control points fall outside the boundaries of the square, unintended defects in visual patterns may occur. Figure 9 provides a visualization of such errors:

Figure 9

Point pattern errors when the control points P₁and P₂ fall outside the square

(a) If P₁ deviates leftward, the central structure vanishes (resembling increased s-value effects).

(b) If P₁ falls below the lower boundary, points randomly overlap.

(c) If P₂ falls above the upper boundary, the entire pattern abnormally enlarges.

(d) If P₂ deviates rightward, the pattern becomes excessively compressed.

(e),(f) If the control lines for P₁ and P₂ intersect or show a kink, the curvature continuity becomes defective.

These results indicate that stable patterns can only be generated when the control points are constrained to fall within the square domain, thereby ensuring predictable visual representations in real-time music visualization.

3. 4. Music Feature Processing

3. 4. 2. Audio Signal Processing

The system uses the Audio Play component of the Grasshopper’s Mosquito plugin to extract real-time waveform data. This component outputs amplitude peak values as a time sequence of double-precision floating-point numbers. Since this format is incompatible with conventional visualization model inputs, data type conversion and normalization are essential.

Figure 10 summarizes the data processing workflow required for the Bézier-based Fibonacci visualization model.

Figure 10

Data processing flow for the Fibonacci-based visualization model

Using this workflow, every port is handled as follows:

P¹, P² Ports: The two-dimensional coordinates of the audio waveform sample are converted to control-point coordinates of a square space.

s Port: Audio Play only accepts single-track input; thus, the default value of center spacing is set at 0.

n Port: Data of the amplitude of the input (increasing or decreasing with time) are modeled with curved values to manage the actual number of points in real time.

w Port: This is set to a fixed value to match the visual range of expression.

Smoothing and Calibration: Sudden changes in the data within very short periods are alleviated to maintain a gradual and continuous development of visual patterns.

3. 4. 4. Component Implementation

These smoothing algorithms were made available in Grasshopper as custom components intended to fit into the data flow and visualization operations. The final data-processing component is presented in Figure 11. With the help of this part, the research developed numerous random curves and graphically experimented with patterns that change in real time according to musical rhythm. As a result, although all curves generated different patterns, they were closely synchronized with the musical beats, which provided the emotional impact of a visual experience.

Figure 11

Final component setup for the music-responsive visual patterns using smoothed audio data

Figure 12 presents an example of music visualization as implemented with our system. By mapping real-time musical data to such parameters as the number of points (n), curvature control points (P₁, P₂), and color variations, we confirmed that the visual patterns organically transform according to the rhythm and emotion of the music (Refer to the accompanying video file for demonstration.).

Figure 12

Output scene of the real-time music-responsive visualization system

4. 1. Technical Performance

Each model was applied in the same Rhino/Grasshopper environment and run with the same hardware configuration (Intel i7-11800H / RTX 3060 / 32GB RAM) to guarantee a fair and repeatable comparison. The compared models were set as follows:

The proposed model (Bézier-weighted dynamic phyllotaxis) was applied as explained in Section 3.

In Baseline 1 (Traditional Vogel Model), we used the formula of Vogel (statics); to allow a performance comparison of the same number of elements rendered over time, we adjusted it to produce the same total number of points (n) per frame as in our dynamic model when driven by the same audio signal, which isolates the computational cost of the layout algorithm itself.

Baseline 2 (Particle System) offers the implementation of a standard CPU-based particle system according to Fonteles et al. (2013), in which the particles are emitted depending on amplitude and move according to simple physics, which is an alternative way of visualizing music.

The metrics below were selected because they are directly related to real-time interactive applications:

Latency is important in live performances because high latency will cause the visuals to lag and be out of sync with the music, thus ruining immersion.

Curvature Range: The wider the range, the better able the visualization is to depict a large range of musical “textures” and “dynamics” (i.e., smooth, gentle curves versus aggressive, twisting lines), and therefore the greater the expressive ability.

Branch Rate is the ability of the system to generate complex geometry in real-time; a higher branch rate is required to generate dense and complex visual patterns that can update in response to complex musical passages without losing frames. It is directly related to one of the fundamental concepts of real-time computer graphics, that the effective management of geometric complexity is key to maintaining interactive frame rates, which has been formalized in the literature on Level of Detail methods (Luebke et al., 2003).

4. 2. Results and Discussion

The results show statistically significant differences across all metrics, as detailed in Table 1.

Table 1

Experimental Results of the technical performance comparison

4. 3. User Perception Study

We adopted a controlled comparative study to appropriately contextualize user feedback in which the proposed dynamic visualization of phyllotaxis was compared to the control condition of a classical visualization of a spectrogram.

Methods:

A total of 50 volunteers took part in this research (11 male, 39 female), and 33 (66%) of the volunteers had experience in music visualization or design. The stimuli were three music samples (slow-tempo, high-rhythm, and real-time input), which were visualized with the help of (A) our proposed method and (B) the spectrogram control. This study employed a within-subjects design counterbalanced in presentation order. The visualizations were assessed using the following statements scored on a 5-point Likert scale (1 = strongly disagree, 5 = strongly agree):

(1) the visualization was accompanied by the music very well; (2) the visual style was a good indication of the emotional variation in the music; (3) the visualization was graphically attractive and appealing to the eye; and (4) the visualization was useful for hearing structural aspects of music (e.g., rhythm and intensity changes).

Results:

To compare the ratings each evaluation item for the two visualization methods, paired-sample t-tests were used. Table 2 shows the average ratings and their statistical significance.

Table 2

Mean user ratings (proposed method vs. spectrogram control)

The suggested technique scored higher than the spectrogram control on three of the four items of evaluation. Music synchronization was significantly better (M = 4.3 vs. 3.5, p <.01), which means that participants felt that the phyllotaxis visualization was more directly related to musical events in time, and the most significant difference was in emotional reflection (M= 4.1 vs. 2.9, p =.001), implying that the organic, nature-inspired patterns were more effective in expressing the emotional aspect of the music. There were also higher ratings of visual engagement and aesthetics (M = 4.4 vs. 3.2, p <.001), and the proposed approach was rated as more visually appealing. The difference in the perception of musical structure, however, was not significant (M = 3.8 vs. 3.7, p =.25), which is understandable since spectrograms are specifically created to analytically represent audio signals. In general, the average rating of the proposed method was much higher (M = 4.15 vs. 3.33, p <.001).